On the randomized solution of initial value problems

On the randomized solution of initial value problems

Recently S. Heinrich and B. Milla presented an order optimal randomized algorithm solving this problem for γ-smooth input data (i.e. γ = r + ρ: the r-th derivatives of f satisfy a ρ-Hölder condition). This algorithm uses function values and values of derivatives of f . In this paper we present an order optimal randomized algorithm for the class of γ-smooth functions that uses only values of f ....

Trigonometrically fitted two-step obrechkoff methods for the numerical solution of periodic initial value problems

In this paper, we present a new two-step trigonometrically fitted symmetric Obrechkoff method. The method is based on the symmetric two-step Obrechkoff method, with eighth algebraic order, high phase-lag order and is constructed to solve IVPs with periodic solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. The numeri...

The randomized complexity of initial value problems

We study the complexity of randomized solution of initial value problems for systems of ordinary differential equations (ODE). The input data are assumed to be γ-smooth (γ = r+% : the r-th derivatives satisfy a %-Hölder condition). Recently, the following almost sharp estimate of the order of the n-th minimal error was given by Kacewicz (Almost optimal solution of initial-value problems by rand...

Numerical solution of fuzzy initial value problems under generalized dierentiability by HPM

Hybrid dynamic iterations for the solution of initial value problems

Many scientific problems are posed as Ordinary Differential Equations (ODEs). A large subset of these are initial value problems, which are typically solved numerically. The solution starts by using a known state-space of the ODE system to determine the state at a subsequent point in time. This process is repeated several times. When the computational demand is high due to large state space, pa...